Explanation:

The linear regression analysis where the Close price of Bitcoin is regressed on the Date reveals several key findings. The formula used in the regression model is Close ~ Date, indicating that the Close price of Bitcoin is modeled as a function of Date. The residuals, which represent the differences between the observed and predicted values, have a wide range, with the minimum residual at -15,114, the first quartile at -7,997, the median at -2,255, the third quartile at 3,065, and the maximum residual at 35,626. This range indicates some degree of variability in the model’s predictions.

The coefficients in the model include an intercept of -221,100 and a Date coefficient of 13.08. The intercept is the estimated value of the Close price when Date is zero. However, in the context of a time series, the intercept mainly serves as a starting point for the regression line. The Date coefficient suggests that, on average, the Bitcoin Close price increases by about $13.08 per unit of time. The standard error for the intercept is 19,390, and for the Date coefficient, it is 1.073. These standard errors measure the average distance that the observed values fall from the regression line, with smaller standard errors indicating more precise estimates.

The t-values and associated p-values for the intercept and Date coefficients are highly significant. The t-value for the intercept is -11.40 with a p-value less than 2e-16, and the t-value for the Date coefficient is 12.19 with a p-value also less than 2e-16. These small p-values indicate that both the intercept and the Date coefficient are statistically significant at conventional significance levels. The residual standard error of the model is 10,430 on 105 degrees of freedom, indicating the typical size of the residuals. A lower value would suggest a better fit of the model to the data.

The multiple R-squared value is 0.586, meaning that approximately 58.6% of the variability in the Close price of Bitcoin can be explained by the Date variable. The adjusted R-squared value is 0.5821, which is a modified version of R-squared that accounts for the number of predictors in the model. The F-statistic for the overall significance of the model is 148.6 on 1 and 105 degrees of freedom, with a p-value less than 2.2e-16, indicating that the model is statistically significant.

In summary, the linear regression model suggests a statistically significant upward trend in Bitcoin prices over time, with the Close price increasing by about $13.08 per unit of time. The model explains approximately 58.6% of the variability in Bitcoin prices, indicating a moderate fit. However, the large residual standard error suggests substantial variability not explained by the model, possibly due to other influencing factors not included in this simple linear model.

Explanation:

The time series plot tracks the monthly closing prices of Bitcoin in US dollars, spanning January 2015 to December 2023 (assuming the data includes December 2023). Each data point on the plot represents an actual monthly closing price, and together they illustrate the price movements over this extended period.

Time Series and Trend: The x-axis acts as our time scale, displaying months with year labels appearing at every January tick mark. The y-axis represents the price of Bitcoin in US dollars. The line superimposed on these data points showcases the general trend in Bitcoin’s closing price over time.

Linear Regression Model: This line represents a statistical model attempting to capture the overall trend in the data. By fitting a straight line through the scatter plot of closing prices, the model offers a simplified representation of the average price movement. The specific equation for this line might be displayed elsewhere but isn’t visible in the image we’re examining.

Key Observations:

Upward Trajectory: By analyzing the time series plot, we can observe a general upward trend. This upward direction is mirrored by the positive slope of the linear regression model. This suggests that, on average, the closing price of Bitcoin has increased throughout the period from January 2015 to November 2023.

Price Fluctuations: Despite the apparent upward trend, the data points themselves reveal significant volatility in Bitcoin’s price. The scatter plot shows these data points spread out around the regression line, indicating that the price has fluctuated considerably from month to month. The linear regression model provides an average trend but doesn’t capture these individual price swings.

Limitations of the Model: It’s crucial to remember that linear regression models are simplifications of real-world phenomena. The price of Bitcoin is influenced by a multitude of complex factors, and a straight line may not perfectly capture the intricate price movements. The deviations of the scatter points from the regression line highlight this limitation.

In conclusion, the graph offers insights into Bitcoin’s price behavior. While the overall trend suggests a general increase over time, the data also demonstrates substantial volatility.

Explanation:

The provided plot depicts residuals from a linear regression model, where residuals represent the differences between observed values and those predicted by the model. The X-axis denotes dates ranging from 2015 to 2024, while the Y-axis measures residuals. In the plot, blue dots (Trace 0) signify individual residual values, while a red line (Trace 1) serves as a trend line, ideally centered around zero.

Interpreting the plot reveals that residuals span from approximately -10,000 to +30,000, showing variability over time. Initially, from 2015 to around 2018, residuals cluster tightly around zero with minimal deviation. Post-2018, however, the spread of residuals noticeably widens. Around 2020, there’s a notable clustering of residuals below zero, while by 2022, variability increases further with some residuals showing notably high positive values.

Analyzing trends, the horizontal red line at zero indicates that residuals should ideally hover around zero, implying a well-fitted model. However, deviations from this line highlight periods where model predictions were less accurate. Potential issues identified include heteroscedasticity, evident from the increasing spread of residuals over time, suggesting that the variance of residuals isn’t constant. Moreover, the model’s fit appears to falter, especially noted by the pronounced spread of residuals in 2022, hinting at possible outliers or influential points impacting accuracy.

In conclusion, the residuals plot points towards limitations in the linear regression model’s ability to predict values consistently across different time periods. Addressing these challenges may require further investigation, potential model refinements, or the incorporation of more sophisticated variables to enhance prediction accuracy.

Explanation:

The provided histogram illustrates the distribution of residuals from a linear regression model, focusing on the differences between observed and predicted values. The X-axis represents the range of residuals, spanning from approximately -20,000 to +40,000, while the Y-axis shows the frequency of residuals within each bin.

Interpreting the histogram reveals that most residuals are centered around zero, indicating that the model’s predictions generally align closely with the observed data. A notable peak at zero suggests a significant number of residuals are very close to this value. However, the distribution exhibits a slight right skew, implying there are some larger positive residuals where the model underestimated the observed values. Fewer residuals appear in the positive range (especially above 20,000), contrasting with a visible number of residuals in the negative range, albeit less extreme.

Moreover, outliers are evident in the positive range of residuals, particularly around 30,000 to 40,000, highlighting instances where the model significantly mispredicted. Despite the majority of residuals being near zero, the histogram’s right skew and the presence of outliers suggest deviations from a perfectly normal distribution. This indicates potential data patterns or anomalies that the linear regression model may not fully capture.

In conclusion, while the linear regression model generally performs well with predictions close to observed values, the histogram suggests areas for improvement. Addressing the right skew, handling outliers, and potentially revising the model or exploring data transformations could enhance its accuracy and better accommodate underlying data nuances.

Explanation:

ACF

This Autocorrelation Function (ACF) plot of residuals from a time series model provides insights into the correlation between residuals and their lagged values. The Y-axis represents the autocorrelation coefficient, measuring this relationship, while the X-axis denotes the number of time steps (lags) between compared residuals. Each vertical bar on the plot illustrates the autocorrelation coefficient for a specific lag, with bar height indicating the strength and direction of correlation. Additionally, blue dashed lines signify confidence intervals typically set at ±2/√n, where n is the sample size, helping identify statistically significant correlations.

Interpreting the plot reveals significant autocorrelations at initial lags, suggesting residual correlation that the model may not have fully captured. This implies potential inadequacies in the model’s ability to account for underlying data patterns. As lag increases, autocorrelations diminish and generally fall within the confidence intervals, indicating that residuals are less correlated at higher lags. Ideally, residuals should exhibit near-zero autocorrelation for all lags if the model is appropriate. Persistent significant autocorrelations imply possible model shortcomings, such as omitted variables or incorrect specifications.

In summary, this ACF plot highlights residual autocorrelation at lower lags, signaling areas where the model could benefit from refinement to better capture data dynamics. Addressing these findings may involve revisiting model assumptions, incorporating additional variables, or exploring alternative model specifications to improve overall predictive accuracy.

PACF

This Partial Autocorrelation Function (PACF) plot of residuals from a time series model provides insights into the partial correlation between residuals and their lagged values, accounting for intervening lags. The Y-axis represents the partial autocorrelation coefficient, which measures this relationship, while the X-axis denotes the number of time steps (lags) between compared residuals. Each vertical bar on the plot illustrates the partial autocorrelation coefficient for a specific lag, with bar height indicating the strength and direction of the partial correlation. Blue dashed lines represent confidence intervals typically set at ±2/√n, where n is the sample size, indicating statistically significant correlations.

Interpreting the plot reveals a significant partial autocorrelation at lag 1, suggesting a direct relationship between residuals and their first lag. This indicates that after accounting for other intervening lags, there remains a notable residual correlation at this lag. For lags beyond 1, the partial autocorrelations generally fall within the confidence intervals, implying no significant residual correlations once the first lag is considered. This pattern suggests that the model adequately captures the relationships between residuals and lagged values beyond the first.

Ideally, residuals should exhibit near-zero partial autocorrelation for all lags if the model is appropriate. The significant partial autocorrelation at lag 1 in this PACF plot suggests that the model might benefit from further refinement to better capture or incorporate a first-order component in its structure. Addressing this finding could involve reassessing model specifications, incorporating additional explanatory variables, or exploring alternative modeling approaches to improve predictive accuracy and account for residual correlations effectively.

Explanation:

Understanding the Plot:

The x-axis represents the theoretical quantiles, or expected values, of a normal distribution. The y-axis represents the quantiles of the residuals calculated from our linear regression model. In an ideal scenario, if the residuals follow a normal distribution, the data points on the plot would closely resemble a straight diagonal line. This would signify that the residuals behave similarly to what we would expect from a normal distribution.

Observations from the Data:

Examining the normal QQ plot we have, it appears that the data points deviate somewhat from the ideal straight diagonal line, particularly at the tails of the distribution (the areas on the far left and right sides of the plot). This suggests a potential departure from normality in the residuals.

Explanation:

The statement “Shapiro-Wilk normality test” followed by the results “data: residuals W = 0.85983, p-value = 1.215e-08” signifies the outcome of a statistical test used to assess whether a set of data, specifically the residuals from a statistical model, adheres to a normal distribution. The Shapiro-Wilk test evaluates the null hypothesis that the data is sampled from a population that follows a normal distribution. In this context, the test was applied to the residuals, which are the differences between observed values and those predicted by the model.

The test statistic, denoted as W, is calculated to quantify how closely the sample resembles a normal distribution. A value of W close to 1 suggests strong conformity to normality. Here, the computed W value is 0.85983, indicating some departure from normality. The critical component of the test is the p-value, which measures the probability of observing the data if the null hypothesis were true (i.e., if the residuals were normally distributed).

In this case, the p-value is very small, specifically 1.215e-08 (or 0.00000001215 in standard notation), which is significantly less than common significance levels like 0.05 or 0.01. This low p-value provides strong evidence against the null hypothesis of normality. Therefore, we reject the null hypothesis and conclude that the residuals do not follow a normal distribution.

The implication of non-normality in residuals is important for interpreting statistical models. Departure from normality can impact the accuracy of statistical inferences and predictions derived from the model. It suggests that the assumptions underlying the model (such as normality of errors) may not be fully satisfied. As a result, further investigation may be warranted to understand the nature of the deviation from normality, potentially requiring adjustments to the model or data transformations to improve its validity and reliability.

Explanation:

The linear regression model aimed to predict Bitcoin’s closing prices using a second-degree polynomial of the numeric date. The model’s coefficients for the intercept, first-degree term, and second-degree term are not statistically significant, with p-values of 0.798, 0.614, and 0.432, respectively. Despite this, the model explains a considerable portion of the variance in the data, as indicated by an R-squared value of 0.5885 and an adjusted R-squared of 0.5806, suggesting that about 58% of the variability in Bitcoin’s closing prices is accounted for by the model. The residual standard error is 10,450, reflecting moderate dispersion of the residuals. The overall model is highly significant, with an F-statistic of 74.36 and a p-value less than 2.2e-16, indicating that the model provides a significant fit to the data as a whole.

Explanation:

ACF

The ACF plot of Bitcoin prices reveals a gradual decline in autocorrelation values as the lag increases, indicating a strong correlation at shorter lags that diminishes over time. Significant autocorrelations are present at many lags, suggesting that past prices notably influence current prices. The slow decay of autocorrelations often points to non-stationarity in the time series, implying that the Bitcoin price series may need differencing to achieve stationarity for effective time series modeling.

PACF

The PACF plot of Bitcoin prices shows a significant spike at lag 1, indicating a strong partial autocorrelation between the current price and the price one period ago. This suggests that the immediate past price has a substantial influence on the current price. Beyond lag 1, the partial autocorrelations drop off and hover close to zero, with none of the subsequent lags showing significant spikes beyond the blue dashed lines (the confidence intervals). This indicates that the effects of previous prices diminish rapidly and do not significantly influence the current price beyond the first lag. The absence of significant lags beyond lag 1 suggests that the time series does not exhibit higher-order autocorrelations, implying that once the effect of the first lag is accounted for, the remaining lags do not contribute additional predictive power.

Comment on the dataset’s nature: The dataset appears to exhibit characteristics of a time series with a strong dependency on its immediate past values. This suggests that the data may follow a random walk or a process where the current value is primarily influenced by its most recent value, with little to no significant autocorrelation beyond the first lag. This nature of the dataset implies that simple autoregressive models, such as AR(1), might be appropriate for modeling and forecasting the Bitcoin prices. Additionally, the lack of higher-order correlations suggests that the series may resemble white noise after accounting for the first lag, indicating that more complex models might not provide substantial additional predictive power.

Explanation:

The Augmented Dickey-Fuller (ADF) test output provides insights into the stationarity of the Bitcoin price series. The test regression indicates a model with a drift component, evaluating the differenced series (z.diff) against lagged values. Descriptive statistics of the residuals, including minimum, quartiles, and maximum values, characterize the variability not explained by the model. Coefficients for the intercept ((Intercept)) and lagged differenced series (z.lag.1) are estimated, with their respective standard errors, t-values, and p-values assessed for significance.

The results show a residual standard error of 4443, indicating the model’s fit and the variability in the dependent variable that remains unexplained. However, the low R-squared (0.01402) and adjusted R-squared (0.004535) suggest the model does not effectively capture the data’s variability, potentially indicating non-stationarity.

The ADF test statistic (-1.2159) is pivotal in determining stationarity, compared against critical values at 1%, 5%, and 10% levels (tau2). With values greater than these critical thresholds, the null hypothesis of a unit root (non-stationarity) cannot be rejected. Additionally, the p-value (0.2268) exceeds conventional significance levels, further supporting the conclusion of non-stationarity.

Explanation:

ACF Plot Analysis: The ACF starts high and declines slowly, indicating a potential AR or ARMA model. Significant spikes at specific lags suggest the presence of autocorrelation at those lags.

PACF Plot Analysis: The PACF has significant spikes at lags 1 and 2, which then drop off, suggesting an AR process. If the PACF cuts off after lag q, it indicates an ARIMA(p, d, q) model with p = 1 or 2, and q = 0.

Perform EACF test to comprehensively test the possible candidate models. Mention the models that you have selected for modeling (select at least 3 models).

Explanation:

Model Selection Criteria:

1. ARIMA(1,1,1):
   
   • \(p = 1\), \(d = 1\), \(q = 1\)
   • This model has significant terms at AR(1) and MA(1), which is a straightforward combination often providing a good fit for various time series.
2. ARIMA(2,1,1):
   
   • \(p = 2\), \(d = 1\), \(q = 1\)
   • This model has significant terms at AR(2) and MA(1). The additional AR term can capture more complexity in the autoregressive part.
3. ARIMA(3,1,2):
   
   • \(p = 3\), \(d = 1\), \(q = 2\)
   • This model has significant terms at AR(3) and MA(2). Including both higher-order AR and MA terms allows for more flexibility in modeling complex patterns in the data.

Explanation of Choices:

1. ARIMA(1,1,1):
   
   • The ARIMA(1,1,1) model is often a good starting point because it balances simplicity with the ability to capture both AR and MA components. It’s chosen because the EACF shows significance in the first AR and MA terms without too much complexity.
2. ARIMA(2,1,1):
   
   • Adding an additional AR term can sometimes capture more subtle dependencies in the data. The EACF shows significance at AR(2) and MA(1), suggesting this model could better fit data with a slightly more complex autoregressive structure.
3. ARIMA(3,1,2):
   
   • This model includes both higher-order AR and MA terms. The presence of significant terms in the EACF table at these positions suggests that this model could better capture complex dynamics and interactions between past values and past forecast errors.

Explanation:

In Model 1, all three coefficients—ar1, ar2, and ma1—are statistically significant. Specifically, ar1 and ar2 are significant at the 0.05 level with p-values of 0.01299 and 0.04218, respectively. The coefficient for ma1 is highly significant, with a p-value less than 2e-16, indicating a very strong contribution to the model.

In Model 2, both ar1 and ma1 coefficients are statistically significant. The ar1 coefficient is significant at the 0.05 level with a p-value of 0.03443, while the ma1 coefficient is again highly significant with a p-value less than 2e-16. This suggests that both terms contribute meaningfully to the model.

Model 3 presents a more mixed picture. The ar1 coefficient is marginally significant with a p-value of 0.060274, which is just above the 0.05 threshold. The ar2 coefficient is not significant, with a p-value of 0.867036, indicating it does not contribute significantly to the model. In contrast, the ar3 coefficient is significant at the 0.05 level with a p-value of 0.024481. Among the moving average terms, ma1 is not significant with a p-value of 0.145019, while ma2 is highly significant at the 0.01 level with a p-value of 0.002796. This indicates that while some terms in Model 3 are important, others do not significantly contribute to the model’s performance.

Explanation:

The accuracy metrics for the ARIMA(2,1,1) and ARIMA(1,1,1) models provide a comparison of their performance in predicting the data. For the ARIMA(2,1,1) model, the Mean Absolute Error (MAE) is 436.1328, indicating the average absolute difference between the predicted values and the actual values. The Mean Squared Error (MSE) is 228802.7, reflecting the average of the squared differences between predicted and actual values, which heavily penalizes larger errors. The Root Mean Squared Error (RMSE), the square root of the MSE, is 478.3333, offering a measure in the same unit as the data, and the Mean Absolute Percentage Error (MAPE) is 2.756649%, indicating the average absolute percentage difference between predicted and actual values.

In comparison, the ARIMA(1,1,1) model has a slightly lower MAE of 425.8315, suggesting marginally better accuracy in terms of average absolute error. The MSE for this model is 235393.7, slightly higher than that of the ARIMA(2,1,1) model, indicating a slight increase in the penalization of larger errors. The RMSE is 485.1739, slightly higher than that of the ARIMA(2,1,1), which also suggests a marginally worse fit. The MAPE for the ARIMA(1,1,1) model is 2.698167%, slightly lower than that of the ARIMA(2,1,1), indicating a marginally better performance in terms of average percentage error.

Overall, both models show similar performance, with the ARIMA(1,1,1) model slightly outperforming the ARIMA(2,1,1) model in terms of MAE and MAPE, but the ARIMA(2,1,1) model having a slightly lower MSE and RMSE. These results suggest that the differences in performance between the two models are minimal, with each model having slight advantages depending on the specific accuracy metric considered.

Explanation:

Residuals Plot

Top Left (ARIMA(2,1,1) Residuals): This plot shows the residuals over time for the ARIMA(2,1,1) model. Residuals are randomly distributed with no discernible pattern, indicating that the model has captured all the information in the data. Top Left (ARIMA(1,1,1) Residuals): The plot displays the residuals over time for the ARIMA(1,1,1) model. Random distribution without a pattern suggests a well-fitted model.

ACF Plots Top Right (ACF of ARIMA(2,1,1) Residuals): The ACF plot for the ARIMA(2,1,1) model residuals shows the correlation of the residuals with their lagged values. For a good model fit, these correlations should be close to zero for all lags which suggests no remaining autocorrelation. Bottom Right (ACF of ARIMA(1,1,1) Residuals): This ACF plot shows the correlation of residuals for the ARIMA(1,1,1) model. Again, residuals ideally exhibits no significant autocorrelation if the model fits well.

PACF Plots Top Right (PACF of ARIMA(2,1,1) Residuals): The PACF plot for the ARIMA(2,1,1) model residuals shows the partial correlation of the residuals with their lagged values. Correlations at specific lags suggest that some information have not been captured by the model. Bottom Right (PACF of ARIMA(1,1,1) Residuals): The PACF plot for the ARIMA(1,1,1) model residuals. Like the ACF plot, partial correlations should be close to zero for a well-fitted model.

Explanation:

ARIMA(2,1,1) Residuals (Left Histogram)

Shape: The histogram shows that most residuals are clustered around zero, indicating that the model’s predictions are generally close to the actual values.

Symmetry: The distribution appears to be slightly skewed to the right, with a longer tail extending towards positive residuals. This suggests that there are some instances where the model’s predictions are higher than the actual values.

Spread: The spread of the residuals seems to be fairly concentrated around zero, with fewer extreme values. This indicates that the majority of the residuals are small, reflecting a good fit of the model to the data.

ARIMA(1,1,1) Residuals (Right Histogram)

Shape: Similar to the ARIMA(2,1,1) model, most residuals are clustered around zero, indicating that this model’s predictions are also generally close to the actual values.

Symmetry: This histogram also shows a slight right skew, with a longer tail extending towards positive residuals, indicating that this model too has instances where predictions are higher than the actual values.

Spread: The residuals appear to be slightly more spread out compared to the ARIMA(2,1,1) model, with a few more extreme values. This suggests that while the model fits well, there are some larger prediction errors compared to the ARIMA(2,1,1) model.

Explanation:

ARIMA(2,1,1)

The ACF plot of the residuals from the ARIMA(2,1,1) model shows that most autocorrelation values fall within the 95% confidence intervals, indicating no significant autocorrelation. A few lags slightly exceed these intervals, suggesting minor autocorrelation, but these deviations are not concerning. The residuals appear to be white noise, meaning the model adequately captures the underlying data. To confirm this, additional diagnostics like PACF plots, QQ plots, and the Ljung-Box test are recommended. Checking residual normality with QQ plots and the Shapiro-Wilk test is also important. If issues are found, consider refining the model.

PACF Plot (2,1,1)

The PACF plot of the ARIMA(2,1,1) model residuals shows most values within the 95% confidence intervals, indicating no significant partial autocorrelation and suggesting the residuals are white noise. Minor deviations are present but not concerning. The model seems to capture the data well. Additional diagnostics and normality checks are recommended to confirm the model’s adequacy. If significant patterns are found, consider refining the model.

ARIMA(1,1,1)

The ACF plot for the residuals of an ARIMA(1,1,1) model shows that most spikes fall within the 95% confidence interval, indicating largely uncorrelated residuals. Minor spikes at lags 3, 5, and 10 suggest some minor autocorrelations. Overall, the ARIMA(1,1,1) model effectively captures the underlying data pattern, with the residuals resembling white noise, although slight refinement might be needed for minor residual patterns.

PACF (1,1,1)

The PACF plot of the ARIMA(1,1,1) model residuals shows that most spikes are within the 95% confidence interval, indicating no significant partial autocorrelation and suggesting the residuals resemble white noise. There are minor spikes around lags 3 and 5, indicating slight partial autocorrelations. Overall, the ARIMA(1,1,1) model appears to fit the data well, but minor residual patterns suggest potential for further model refinement.

Explanation:

ARIMA(2,1,1) Model Residuals For the ARIMA(2,1,1) model, the Shapiro-Wilk normality test results indicates a deviation from normality in the residuals. The W statistic is 0.84254, which is substantially less than 1, suggesting that the residuals do not closely follow a normal distribution. The p-value is 3e-09, an extremely small value, which strongly rejects the null hypothesis that the residuals are normally distributed. This outcome implies that the ARIMA(2,1,1) model has not fully captured all the underlying patterns in the data, potentially leaving some structure or information unaccounted for.

ARIMA(1,1,1) Model Residuals Similarly, the ARIMA(1,1,1) model’s residuals also shows a deviation from normality based on the Shapiro-Wilk normality test. The W statistic for this model is 0.84726, again indicating that the residuals are not close to a normal distribution. The p-value is 4.489e-09, which is extremely small and strongly rejects the null hypothesis of normality. This result suggests that, like the ARIMA(2,1,1) model, the ARIMA(1,1,1) model has not fully captured all the variability and patterns in the data, indicating that further refinement or consideration of alternative models may be needed to achieve a better fit.